Need a 2 x 2 matrix who's square is zero

In summary, the conversation discusses finding a 2 x 2 matrix with a square of 0 and determinant of 0. The speaker presents their attempt at a solution and discusses the possibility of finding such a matrix with all nonzero entries. The conversation ends with the speaker expressing their understanding and gratitude for the help.
  • #1
Irks23
3
0

Homework Statement



If the product of two matrices is zero, it is not necessary that either one be zero. In particular, show that a 2 x 2 matrix who's square is zero may be written in two parameters a and b, and find the general form of the matrix. Verify that it's determinant is 0.

The Attempt at a Solution



So I tried to find a 2 x 2 matrix who's square is 0 and determinant is 0. The best I could come up with was:
A iA
iA A
which squares to:
0 2iA^2
2iA^2 0
but only the real part is 0, not the imaginary part, and the determinant is not zero. I'm not seeing at this point how any matrix's square could be zero except the 0 matrix, am I interpreting the question wrong?
 
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  • #2
If A2=0, what can you say about its eigenvalues?

Then if it's, say, upper triangular...
 
  • #3
A B
C D
if its upper triangular, in ABCD format above with C being 0, when you square the matrix, you get A^2+BC=A^2 for the first element, which only is 0 if A is zero...

I really don't know what A^2 = 0 tells about the eigenvalues...
 
  • #4
I can find 3 families, but I don't think you can all write them in one general form

since a=0 didn't work, try b= or c=0 to get two of the families

If the matrix has all nonzero entries, you can do something with
ab + bd = 0 and ac + cd = 0
 
  • #5
got it, thank you
 

1. What is a 2 x 2 matrix?

A 2 x 2 matrix is a rectangular array of numbers arranged in two rows and two columns. It is commonly used in mathematics and science to represent data or equations.

2. How do you square a matrix?

To square a matrix, you multiply the matrix by itself. In other words, you multiply each element in the matrix by each element in the same position in the second matrix and then add the products together. This will result in a new matrix with the same dimensions as the original.

3. What does it mean for a matrix to have a square of zero?

If a matrix has a square of zero, it means that when you square the matrix, the resulting matrix will have all zero elements. In other words, the original matrix multiplied by itself will result in a matrix of all zeros.

4. Can a 2 x 2 matrix have a square of zero?

Yes, a 2 x 2 matrix can have a square of zero. In fact, any square matrix (a matrix with the same number of rows and columns) can have a square of zero if all of its elements are zero.

5. What are the applications of a 2 x 2 matrix with a square of zero?

A 2 x 2 matrix with a square of zero has various applications in mathematics and science. It can be used to solve systems of equations, represent transformations in geometry, and study linear transformations. In physics, it can be used to represent the moment of inertia tensor and study rotational motion.

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